Your sentence is correct, but you are missing the article 'a' before birthday present. Gift and present are synonyms and can be used interchangeably. The only difference is a gift can also apply to non-material things. For example, you could be given the gift of time or insight. You can be gifted with talents or skills. In your case, they mean the same thing. I hope this answer helps. If you are interested in taking English lessons, I am currently accepting new students and would be happy to work with you!

Jazz is right about the missing "a." Your original sentence is correct, other than that. You don't need to switch "gift" and "present." I think it sounds better the way you wrote it first: "I prefer a gift rather than money as a birthday present." It's true that a "gift" is a more general term, and can refer to non-material things in some contexts, but in this context, it's obvious that you mean a material gift. And the phrase "birthday present" sounds more natural than "birthday gift," so I think the sentence is a bit clearer if you use this term.

Missing 'a'. However I question a different issue to the one you raise. The use of 'rather than' In your example I'd use 'to' rather than 'rather than' !哈哈 I prefer a gift to money as a bd prezzie BUT I prefer to be given a gift rather than money. Why 'rather than'? Haven't worked out why yet! 哎呀 I suspect the reason relates to the presence or absence of the preceding verb phrase